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Theorem undir 3702
Description: Distributive law for union over intersection. Theorem 29 of [Suppes] p. 27. (Contributed by NM, 30-Sep-2002.)
Assertion
Ref Expression
undir  |-  ( ( A  i^i  B )  u.  C )  =  ( ( A  u.  C )  i^i  ( B  u.  C )
)

Proof of Theorem undir
StepHypRef Expression
1 undi 3700 . 2  |-  ( C  u.  ( A  i^i  B ) )  =  ( ( C  u.  A
)  i^i  ( C  u.  B ) )
2 uncom 3603 . 2  |-  ( ( A  i^i  B )  u.  C )  =  ( C  u.  ( A  i^i  B ) )
3 uncom 3603 . . 3  |-  ( A  u.  C )  =  ( C  u.  A
)
4 uncom 3603 . . 3  |-  ( B  u.  C )  =  ( C  u.  B
)
53, 4ineq12i 3653 . 2  |-  ( ( A  u.  C )  i^i  ( B  u.  C ) )  =  ( ( C  u.  A )  i^i  ( C  u.  B )
)
61, 2, 53eqtr4i 2491 1  |-  ( ( A  i^i  B )  u.  C )  =  ( ( A  u.  C )  i^i  ( B  u.  C )
)
Colors of variables: wff setvar class
Syntax hints:    = wceq 1370    u. cun 3429    i^i cin 3430
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1592  ax-4 1603  ax-5 1671  ax-6 1710  ax-7 1730  ax-10 1777  ax-11 1782  ax-12 1794  ax-13 1954  ax-ext 2431
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-tru 1373  df-ex 1588  df-nf 1591  df-sb 1703  df-clab 2438  df-cleq 2444  df-clel 2447  df-nfc 2602  df-v 3074  df-un 3436  df-in 3438
This theorem is referenced by:  undif1  3857  dfif4  3907  dfif5  3908  bwth  19140
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